A Mathematica Tutorial

Transcription

1 A Mathematica Tutorial -3-8 This is a brief introduction to Mathematica, the symbolic mathematics program. This tutorial is generic, in the sense that you can use it no matter what kind of computer you have. However, this means that it doesn t discuss features which are specific to your version of Mathematica. Hence, if you can find a tutorial designed for your system, you might find it more useful. This tutorial assumes that you know how to perform basic operations with your computer. If not, you should consult the documentation that came with your machine. I ll start with basic arithmetic. Here is the Mathematica command to compute 3 : 3^ Type it in, then execute it by pressing Shift-Enter. (That is, hold down one of the Shift keys, press Enter, then release both keys.) (If you are using the Macintosh version or the Microsoft Windows version, move the mouse until your cursor turns into a horizontal I-beam. Click the mouse; a horizontal line appears. When you start typing, your input should replace the horizontal line. When you are finished typing, press Shift-Enter to execute the command.) The long number that appeared is 3. Notice that Mathematica put In[]:= next to your input and Out[]= next to your output. Mathematica labels all of its input and output in this way so that you have an easy way to refer to things you ve done earlier. Notice that I m using a different font and typeface for Mathematica commands. A Mathematica command will always look like this: (* Here is a Mathematica comment. *) That way you can tell it apart from normal text. As you go through the rest of the tutorial, my explanations will be interspersed with Mathematica commands for you to try. You should execute the commands as you read to see how they work. Here are some things to pay particular attention to: Spaces (or the absence of spaces) can be important! For example, xy and x y are not the same! The first is the -letter name of a variable, i.e. a single word. The second is x times y. If you want x multiplied by y you can either leave a space between x and y or write a multiplication sign like this: x y. Certain letters must be capitalized. For example, most of Mathematica s built-in functions start with a capital letter, and some have capital letters in the middle. Plot3D is correct; plot3d is not. Pay attention to punctuation. For example, (, [, and { mean different things.. More arithmetic. Here is how to use Mathematica to perform addition, subtraction multiplication, and division. Execute the commands below; for practice, make up a few yourself * / 9. Numerical results. Of course, you can use Mathematica to compute transcendental functions. Execute the next line:

2 Sin[3] The result isn t very enlightening. To express a result in approximate (decimal) form, use the N[] function: N[Sin[3]] Mathematica (like most mathematicians) reads angles in radians unless you tell it otherwise. So Sin[3] means the sine of 3 radians. The names of Mathematica s built-in functions, like Sin, always start with capitals. Some even have capitals in the middle of their names (like ParametricPlot ). You must have exactly the right letters capitalized, or Mathematica won t know what you mean. Notice that when you want to apply a function to something, you must use square brackets: [ and ], not parentheses. That is, instead of writing y = f(x) the way you do in calculus, you would need to write f[x] in Mathematica. So you write Sin[3], not Sin(3). Suppose you want the sine of 3 to 5 places. Execute the next command: N[%, 5] The 5 tells N[] to approximate the result to 5 places. What about the %? This is shorthand for the previous output. It is useful when you want to avoid retyping a long formula, or when you want to use the last result in your next computation. Execute the following commands to compute some other transcendental functions: N[ArcTan[7]] Notice that the function is ArcTan, not Arctan! Mathematica represents the constant e = with a capital E: E. So here s the approximate value of e : N[E^] Note that Log is the natural log function: N[Log[]] 3. Symbolic algebra. Mathematica really shines as a tool for symbolic manipulation. I ll discuss a few of its capabilities below. Execute the commands as you go. Here is a binomial expansion: Expand[(x + y)^] The Expand function is good for multiplying stuff out. You can also use Mathematica to put things together. For example, here is how to combine fractions over a common denominator: Together[/(x + ) - 5/(x^ + )] You can also take a fraction apart (i.e. obtain its partial fractions decomposition). This will be familiar to people who have seen it used as an integration technique. Apart[%] Mathematica can also simplify expressions: Simplify[x^/(x - ) - /(x - )]

3 Mathematica combined the fractions over a common denominator, then noticed that the numerator was divisible by the denominator and performed the cancellation. Mathematica makes reasonable assumptions about how you d like a simplification performed. However, the program can t anticipate every human intention, and sometimes you will have to play around to get answers in the form that you want. Sometimes, it s even necessary to teach Mathematica simplification rules that it doesn t know! If you re doing serious work with Mathematica, you should refer to the manual for other functions and techniques that are useful in performing algebraic manipulations.. Solving equations. You can use Mathematica to solve equations, or systems of equations. For example: Solve[{x + y ==, x - y == }, {x, y}] In some cases, Mathematica can t obtain an exact solution: Solve[x^5 + x == 7, x] You can use N[] to obtain a numerical approximation to the solution: N[%] Here s how to compute d x dx sinx+cosx : D[x/(Sin[x] + Cos[x]), x] 5. Calculus. You can also compute higher-order derivatives. This is the 3-rd derivative of D[x/(Sin[x] + Cos[x]), {x, 3}] You can also compute partial derivatives: D[(x^ + y^) E^(-x^ - y^), x] Mathematica has a sizeable repertoire of integration rules: Integrate[Cos[x]^, x] It is also possible to teach Mathematica rules that it doesn t know. x sinx+cosx : If Mathematica can t evaluate a definite integral in closed form, you can use N[] as usual to obtain a numerical approximation: Integrate[E^(x^), {x,, }] N[%] 6. -dimensional Graphs. This is how to draw the graph of y = xsin x for 3 x 3: Plot[x Sin[/x], {x, -3, 3}] 3

4 Parametric plots are also very simple: ParametricPlot[{Cos[t]^3, Sin[t]^3}, {t, - Pi, Pi}] The aspect ratio defaults to /GoldenRatio; if you want better scaling, you can set the aspect ratio yourself, or have Mathematica do it automatically: ParametricPlot[{Cos[t]^3, Sin[t]^3}, {t, - Pi, Pi}, AspectRatio->Automatic] There are many other options for the graphics functions; see the Mathematica manual for details dimensional Graphs. Here is the graph of z = sinxsiny, for x 3π and y 3π:

5 Plot3D[Sin[x] Sin[y], {x,, 3 Pi}, {y,, 3 Pi}] Many surfaces can t be represented as graphs of functions. Suppose I want to plot x = (+cosu)cosv, y = +sinu, z = (+cosu)sinv. Mathematica s ParametricPlot3D[] function plots parametrized surfaces. Notice that the three surface expressions are enclosed in square brackets: ParametricPlot3D[{( + Cos[u]) Cos[v], + Sin[u], ( + Cos[u]) Sin[v]}, {u,, Pi}, {v,, Pi}] Note: I split the last command into two lines so it would fit on this page, but you aren t required to do that when you type it in. Mathematica will often break lines for you automatically as you type, or you can press Enter to make a line break remember that Shift-Enter actually executes a command. c 8 by Bruce Ikenaga 5